One side of a triangle is 5 cm and the other side is 10 cm and its perimeter is P cm, where P is an integer. The least and the greatest possible values of P are, respectively.

Options

A. 22 and 27
B. 19 and 29
C. 20 and 28
D. 21 and 29

Answer

To find the least and greatest possible values of the perimeter \( P \) of a triangle with sides of lengths 5 cm and 10 cm, we can follow these steps:

### Step 1: Understand the properties of a triangle
In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the triangle inequality theorem.

### Step 2: Define the sides of the triangle
Let the sides of the triangle be:
- \( a = 5 \) cm
- \( b = 10 \) cm
- \( c \) = length of the third side (unknown)

### Step 3: Apply the triangle inequality theorem
According to the triangle inequality theorem, we have the following inequalities:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Substituting the known values into these inequalities:

1. \( 5 + 10 > c \) → \( 15 > c \) → \( c < 15 \)
2. \( 5 + c > 10 \) → \( c > 5 \)
3. \( 10 + c > 5 \) → This inequality is always satisfied since \( c \) is positive.

### Step 4: Determine the range for \( c \)
From the inequalities derived, we have:
- \( 5 < c < 15 \)

This means that \( c \) can take any value greater than 5 cm and less than 15 cm.

### Step 5: Calculate the perimeter \( P \)
The perimeter \( P \) of the triangle is given by:
\[ P = a + b + c = 5 + 10 + c = 15 + c \]

### Step 6: Find the least and greatest possible values of \( P \)
- The least value of \( c \) is just above 5, so we can take \( c = 5 + \epsilon \) (where \( \epsilon \) is a very small positive number). Thus, the least value of \( P \) approaches:
\[ P_{\text{min}} = 15 + (5 + \epsilon) \approx 20 \]
Since \( P \) must be an integer, the least integer value of \( P \) is 20.

- The greatest value of \( c \) is just below 15, so we can take \( c = 15 - \epsilon \). Thus, the greatest value of \( P \) approaches:
\[ P_{\text{max}} = 15 + (15 - \epsilon) \approx 30 \]
Since \( P \) must be an integer, the greatest integer value of \( P \) is 29.

### Conclusion
The least and greatest possible values of \( P \) are:
- Least possible value of \( P = 20 \)
- Greatest possible value of \( P = 29 \)